Question

In an examination, 52% candidates failed in English and 42% failed in Mathematics. If 17% candidates failed in both English and Mathematics, what percentage of candidates passed in both the subjects?
(a) 18%
(b) 21%
(c) 23%
(d) 25%

Answer 1

Hint: We will fix the number of candidates. Then we will calculate the number of candidates who failed in English, the number of candidates who failed in Mathematics, and the number of candidates who failed in both subjects using the given information. After that, we will calculate the number of candidates who have failed in at least one subject. Then we will find the number of candidates who passed in both subjects.

Complete step by step answer:
The information provided to us is in the form of percentages. So, for convenience, let us assume that the total number of candidates is 100. We know that $ x\% $ of 100 is $ x $ . Now, 52% of candidates failed in English. Therefore, there are 52 candidates who have failed in English. Similarly, there are 42 candidates who have failed in Mathematics. We are given that there are 17% of candidates failed in both the subjects. So, there are 17 candidates who failed in both subjects.
We can calculate the number of candidates who have failed in at least one subject in the following manner,
 $ \begin{align}
  & \text{number of candidates who failed at least one subject}=\text{number of candidates failed in English}+ \\
 & \text{number of candidates failed in Mathematics}-\text{number of candidates failed in both subjects} \\
\end{align} $

Therefore, we get
 $ \begin{align}
  & \text{number of candidates who failed at least one subject}=52+42-17 \\
 & \therefore \text{number of candidates who failed at least one subject}=77 \\
\end{align} $
The number of candidates who have passed in both subjects are the candidates remaining after subtracting the number of candidates who have failed in at least one subject from the total number of candidates.
Hence, we have
 $ \begin{align}
  & \text{number of candidates passed in both subjects}=100-77 \\
 & \therefore \text{number of candidates passed in both subjects}=23 \\
\end{align} $
Since, the total number of candidates is assumed to be 100, the percentage of candidates that passed in both the subjects is 23%. Hence, the correct option is (c).

Note:
 We subtracted the number of candidates who failed both the subjects because we had counted them twice. Once, in the number of candidates who failed English and the second time in the number of candidates who failed Mathematics. We can also represent these sets in the form of Venn diagrams, which is a pictorial representation of the information.